Number of spanning clusters at the high-dimensional percolation thresholds

Abstract

A scaling theory is used to derive the dependence of the average number <k> of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6, and vary as log L at d=6. The predictions for d>6 depend on the boundary conditions, and the results there may vary between L^{d-6} and L^0. While simulations in six dimensions are consistent with this prediction (after including corrections of order loglog L), in five dimensions the average number of spanning clusters still increases as log L even up to L = 201. However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L)=1+const/L, indicating that for sufficiently large L the average <k> will approach a finite value: a fit of the 5D multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.

8 pages, 11 figures. Final version to appear on Physical Review E

References in corpus (6)

• On the Number of Incipient Spanning Clusters
• Simulation of percolation on massively-parallel computers
• Lattice ${\varphi }^4$ theory of finite-size effects above the upper critical dimension
• On the Aizenman exponent in critical percolation
• Asynchronously parallelised percolation on distributed machines
• Percolation in high dimensions is not understood

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• High-temperature series expansions for the $q$-state Potts model on a hypercubic lattice and critical properties of percolation